Normal Distributions and Rescaling

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# Normal Distributions and Rescaling
### S. Mason Garrison

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<div class="my-footer">
<span>
<a href="https://DataScience4Psych.github.io/DataScience4Psych/" target="_blank">Methods in Psychological Research</a>
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# Normal Distribution

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## Normal Distribution

Def: a particular bell-shaped curve that has the following mathematical properties

`$f(x)= \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^{2}}$`

- Formula has two parameters
    - `$\mu$`
    - `$\sigma$`
- The standard normal `$(\mu=0; \sigma=1)$` simplifies the equation

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# Standard Normal with multiple means

- The mean is located at the center of the symmetric curve and is the same as the median. 
- Changing `$\mu$` without changing `$\sigma$` moves the Normal curve along the horizontal axis without changing its variability.

.small[
<img src="data:image/png;base64,#rnorms_files/figure-html/norm-1.png" width="55%" style="display: block; margin: auto;" />
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.small[

```r
#### Normal Distribution

# Display the normal distributions with various means

x <- seq(-80, 80, length=1000)
hx <- dnorm(x)
colors <- c("red", "blue","green", "green", "gold", "black")

plot(x, hx, type="l", lty=2, xlab="x value",
		 ylab="Density", main="Comparison of Normal Distributions",xlim=c(-5, 7))
location<-c(2,4,-2)
for (i in 1:3){
	lines(x, dnorm(x,mean=location[i]), lwd=1, col=colors[i])
}
```

<img src="data:image/png;base64,#rnorms_files/figure-html/unnamed-chunk-2-1.png" width="40%" style="display: block; margin: auto;" />
]

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# Plots of the Standard Normal with multiple standard deviations
.pull-left[
- The standard deviation `$\sigma$` controls the variability of a Normal curve. 
-When the standard deviation is larger, the area under the normal curve is less concentrated about the mean.
- The standard deviation is the distance from the center to the change-of-curvature points on either side.
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.pull-right.small[
<img src="data:image/png;base64,#rnorms_files/figure-html/zsd-1.png" width="90%" style="display: block; margin: auto;" />
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---

.small[

```r
# Display the normal distributions with various standard deviations
plot(x, hx, type="l", lty=2, xlab="x value",
		 ylab="Density", main="Comparison of normal Distributions",xlim=c(-10, 10))

for (i in c(.5,2,4,6)){
	lines(x, dnorm(x,sd=i), lwd=1, col=colors[i])
}
```

<img src="data:image/png;base64,#rnorms_files/figure-html/unnamed-chunk-3-1.png" width="40%" style="display: block; margin: auto;" />
]
---

# Normal Distribution

.pull-left[
- In the Normal distribution, with mean `$\mu$`  and standard deviation `$\sigma$`:
    - approximately 68\% of the observations fall within 1 `$\sigma$` of `$\mu$`
    - approximately 95\% of the observations fall within 2 `$\sigma$` of `$\mu$`
    - approximately 99.7\% of the observations fall within 3 `$\sigma$` of `$\mu$`
- This property is sometimes called: The 68-95-99.7 Rule
]
.pull-right[
<img src="data:image/png;base64,#../img/normal.png" width="95%" style="display: block; margin: auto;" />
]

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<img src="data:image/png;base64,#../img/normal.png" width="70%" style="display: block; margin: auto;" />
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# Worked Example

.pull-left[
- The distribution of Iowa Test of Basic Skills (ITBS) vocabulary scores for seventh-grade students in Gary, Indiana, is close to Normal. 
- Suppose the distribution is N(6.84, 1.55).
]
--
.pull-right[
- Sketch the Normal density curve for this distribution.
- What percent of ITBS scores is between 3.74 and 9.94?
- What percent of the scores is above 5.29?
]
--
<img src="data:image/png;base64,#../img/norms.png" width="55%" style="display: block; margin: auto;" />

---
<br>
<img src="data:image/png;base64,#../img/norms.png" width="100%" style="display: block; margin: auto;" />

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# Standard Normal

- Normal is a model of the real world
- Not exact, but it is a facile model for many things
    - Physical features
    - Psychological features
    - Performance measures
--
    - Not all variables are normal
        - Skewed variables (e.g. income)
        - Any count variable (number of kids, mistakes on an exam)
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# Real World Data
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- Many variables follow this distribution ( but not all)
- I have plotted histograms of data,
    - we have already used in this class
    - overlaid with the standard normal.
]
--
.small.pull-right[
<img src="data:image/png;base64,#rnorms_files/figure-html/example-1.png" width="90%" style="display: block; margin: auto;" />
]
---

# Height of Children (Galton dataset)
.small[

```r
library(HistData)
library(ggplot2)

ggplot(Galton, aes(x = child)) + 
	geom_histogram(fill = "red") + 
	stat_function(
		fun = function(x, mean, sd, n){
			n * dnorm(x = x, mean = mean, sd = sd)
		}, 
		args = with(Galton, c(mean = mean(child), sd = sd(child), n
= length(child)))
	) + 
	scale_x_continuous("Heights of Children") 
```

<img src="data:image/png;base64,#rnorms_files/figure-html/unnamed-chunk-8-1.png" width="40%" style="display: block; margin: auto;" />
]

---

# IMBD movie ratings (movies dataset)
.pull-left.midi[

```r
library(ggplot2movies)
data(movies)

plotted_dataset=movies
plotted_dataset$variablex=movies$rating
variablex=movies$rating

ggmovie=ggplot(plotted_dataset, 
               aes(x =variablex)) + 
	geom_histogram(fill = "blue") +
	geom_freqpoly(aes(
	  x=rnorm(length(variablex))*sd(variablex)+mean(variablex)),
	              fill = "black")  +
	scale_x_continuous("IMBD Movie Ratings")  
```
]
.pull-right[
<img src="data:image/png;base64,#rnorms_files/figure-html/unnamed-chunk-9-1.png" width="90%" style="display: block; margin: auto;" />
]
---

# Temperature in Nottingham (nottem dataset)

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# Standard Normal
- Normal distribution tricks
    - Symmetric
    - 50% of area above zero
    - Total proportion is 1.0 (or 100%)
    
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# Area under the Normal Distribution

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# Wrapping Up...

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# Rescaling

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# Rescaling

- All Normal distributions are the same 
  - if we measure in units of size `$\sigma$` from the mean `$\mu$` as center.
- We can convert any variable into the same metric as the standard normal
- Changing to these units is called standardizing or rescaling.

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# Converting Formulas

.pull-left[
- Statistics Sample
- `$z_{i}$` = `$\frac{x_{i}-\bar{x}}{s}$`
]

--
.pull-right[
- Population
- `$z_{i}$` = `$\frac{x_{i}-\mu}{\sigma}$`
]

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# Merits
.pull-left[
- Advantages
    - Allows us to compare scores on a common metric 
    - Origin is 0. The mean
        - The units are 1, the standard deviation
        - '+' values above the mean
        - '-' values below the mean
]

--
.pull-right[
- We can compare across measurement scales
    - Shape of the distribution does NOT CHANGE
- We can go from z-scores to raw scores
]

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# Demo

```r
library(ggplot2movies)
# Rescaling
variable<-movies$rating
scale(variable)[1:10]
```

```
##  [1]  0.30079877  0.04323788  1.45982279  1.45982279 -1.63090793
##  [6] -1.05139592 -0.40749368  0.49396944  0.42957922  0.04323788
```

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# Demo

.pull-left[

```r
plot(density(variable)) # no scaling
```

<img src="data:image/png;base64,#rnorms_files/figure-html/unnamed-chunk-12-1.png" width="90%" style="display: block; margin: auto;" />
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.pull-right[

```r
plot(density(scale(variable))) # with scaling
```

<img src="data:image/png;base64,#rnorms_files/figure-html/unnamed-chunk-13-1.png" width="90%" style="display: block; margin: auto;" />
]

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# Worked Z-Score Problem

- Here are the IQ test scores of 31 7th-grade girls in a Midwest school district. 
<br>

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# Worked Z-Score Problem

A) We expect IQ scores to be approximately Normal.
--

- Make a stem plot to check that there are no major departures from normality.

<img src="data:image/png;base64,#../img/iqstem.png" width="65%" style="display: block; margin: auto;" />
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# Worked Z-Score Problem

B) Find the mean and standard deviation
--

- Mean =105.84 = `$\sum \frac{X_{i}}{n}$` = 3281/31
- SD = 14.27 =  `$s^{2}$` = `$\frac{\sum^{n}_{i=1}(x_{i}-\bar{x})^{2}}{n-1}$` =  `$s^{2}$` = `$\frac{\sum^{n}_{i=1}(x_{i}-105.84)^{2}}{30}$`

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# Worked Z-Score Problem

C) What proportion of scores are within one standard deviation of the mean?
- One SD above mean = 105.84+14.27 = 120.11
- One SD below mean = 105.84 -14.27 = 91.57
- 23/31 = .74
--
<img src="data:image/png;base64,#../img/workz.png" width="65%" style="display: block; margin: auto;" />

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# Worked Z-score Problem

B) What proportion of scores are within TWO standard deviations of the mean?
- TWO SD above mean = 105.84+2(14.27) = 134.38
- TWO SD below mean = 105.84 - 2(14.27) = 77.3
- 29/31 = .935

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# Worked Z-score Problem
B) What would these proportions be in an exactly Normal distribution?
-  +/- One SD
- Continued in Power Point...

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# Wrapping Up...